Could someone please check my work and let me know if I've correctly used all of the included theorems that are referenced in the proof

Please state all definitions and theorems that you will need: Theorem 5.1.13 Let f: D -+ R and g: D -+ R and let c be an accumulation pint of D . If lim f(x) = L, lim g(z) = M , and k E R , then lim (f + 9)(z) = L+ M. lim (fg)(x) = LM , and lim (kf)(x) = KL . Furthermore, if g(x) * 0 for all at E D and M # 0 , then lim (# ) (2) = M. Theorem 4.2.1 Suppose that (S ) and (tn) are convergent sequences with lim s,, = s and lim tn = t . Then (a) lim (8n + tn) = s+ t (b) lim (ksn) = ks and lim (k + 8) = k + 8 , for any k E R (c) lim (sn . tn) = st (d) lim Sn = 7 , provided that tn # 0 for all n and t # 0 .... Theorem 5.2.2 Let f: D -+ R and let c E D . Then the following three conditions are equivalent. (a) f is continuous at c (b) If (In) is any sequence in D such that (n) converges to c, then lim f(an) = f(c) (c) For every neighborhood V of f(c) there exists a neighborhood U of c such that f(UnD) CV. Furthermore, if c is an accumulation point of D , then the above are all equivalent to (d) f has a limit at c and lim f(a) = f(c) . Theorem 5.3.6 (Intermediate Value Theorem) Suppose that f: [a, b] - R is continuous. Then f has the intermediate value property on [a, b] . That is, if k is any value between f(a) and f(b) Li.e., f(a) a but f(b) 0 and f(b) - b 0 and F(b) = f(b) - 6